Victorious warriors win first and then go to war, while defeated warriors go to war first and then seek to win.

Yes. The contrast is in The Art of War, chapter IV, section 15, and the sentence people share is a sound modern translation of it. It is not what Lionel Giles wrote in 1910. The Chinese is 胜兵先胜而后求战，败兵先战而后求胜. Giles says that in war the victorious strategist only seeks battle after the victory has been won, whereas he who is destined to defeat first fights and afterwards looks for victory. The order of the two halves is the whole point of the section.

The modern wording says "warriors" and "go to war", where Giles says "strategist" and "seeks battle". The sequence does not change. One side has already won before the fighting starts. The other fights first and goes looking for a win afterwards. Cite IV.15. If you need a public-domain sentence, use Giles and name the year. If you prefer the shorter modern line, present it as a modern translation of the same section. Either way, the result is settled before the battle, or it is not this passage.